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Equalized Odds (Fairness Analysis)

Equalized Odds detects shortcuts by measuring True Positive Rate (TPR) and False Positive Rate (FPR) disparities across protected groups.

Based on Hardt et al. (2016), this method checks whether a classifier trained on embeddings satisfies the equalized odds criterion: equal TPR and FPR across all demographic groups.

Large gaps indicate that the embeddings encode group-dependent shortcuts that cause differential performance.

How It Works

The Equalized Odds detector:

  1. Trains a lightweight classifier (LogisticRegression by default) on embeddings to predict task labels
  2. Computes per-group metrics: TPR and FPR for each protected group
  3. Calculates gaps: gap = max(rate) - min(rate) across groups
  4. Assesses risk based on gap thresholds
graph TD
    A[Embeddings + Labels + Groups] --> B[Train Classifier]
    B --> C[Predictions]
    C --> D[Per-Group TPR/FPR]
    D --> E[Compute Gaps]
    E --> F{Gap >= Threshold?}
    F -->|Yes| G[Shortcut Detected]
    F -->|No| H[No Shortcut]
Loading

Key insight: If TPR or FPR differs significantly across groups, the model treats groups unequally for the same ground truth, suggesting shortcut reliance.

Basic Usage

Via the Unified API (Recommended)

from shortcut_detect import ShortcutDetector
import numpy as np

# Precomputed embeddings
embeddings = np.load("embeddings.npy")
labels = np.load("labels.npy")  # Binary labels required
group_labels = np.load("groups.npy")  # Required

detector = ShortcutDetector(methods=["equalized_odds"])
detector.fit(embeddings, labels, group_labels=group_labels)

print(detector.summary())

Standalone Usage

from shortcut_detect.fairness import EqualizedOddsDetector

detector = EqualizedOddsDetector(
    tpr_gap_threshold=0.1,
    fpr_gap_threshold=0.1,
    min_group_size=10,
)

detector.fit(embeddings, labels, group_labels)

print(f"TPR Gap: {detector.tpr_gap_:.3f}")
print(f"FPR Gap: {detector.fpr_gap_:.3f}")
print(f"Risk Level: {detector.report_.risk_level}")

Parameters

Parameter Type Default Description
estimator Estimator LogisticRegression Classifier to train on embeddings
min_group_size int 10 Minimum samples per group (smaller groups get NaN)
tpr_gap_threshold float 0.1 Threshold for flagging TPR disparity
fpr_gap_threshold float 0.1 Threshold for flagging FPR disparity

Outputs

Report Structure

results_["equalized_odds"]["report"] contains an EqualizedOddsReport:

Field Type Description
group_metrics dict Per-group TPR, FPR, support, and confusion matrix counts
tpr_gap float Max TPR - Min TPR across groups
fpr_gap float Max FPR - Min FPR across groups
overall_accuracy float Classifier accuracy on all samples
risk_level str "low", "moderate", or "high"
notes str Human-readable interpretation
reference str "Hardt et al. 2016"

Per-Group Metrics

For each group, group_metrics[group_id] contains:

Metric Description
tpr True Positive Rate = TP / (TP + FN)
fpr False Positive Rate = FP / (FP + TN)
support Number of samples in the group
tp, fp, tn, fn Confusion matrix counts

Interpretation

Risk Assessment

Risk Level Condition Interpretation
Low Both gaps < threshold Equalized odds approximately satisfied
Moderate Any gap >= threshold Noticeable disparity in TPR or FPR
High Any gap >= 2x threshold Large disparity, strong evidence of shortcuts

What the Metrics Mean

Pattern Interpretation
High TPR gap Model catches positives better for some groups than others
High FPR gap Model makes more false alarms for some groups
Both gaps high Systematic unfairness across both error types
Low accuracy + low gaps Fair but poorly performing model

Rule of thumb

If TPR or FPR differs by more than 10% across groups, the embeddings likely encode shortcuts correlated with group membership.

Example with Synthetic Data

from shortcut_detect import ShortcutDetector
import numpy as np

np.random.seed(42)

# Create embeddings where group 0 has stronger signal
n = 500
embeddings = np.random.randn(n, 20)
labels = (embeddings[:, 0] > 0).astype(int)
groups = np.array([0] * 250 + [1] * 250)

# Add shortcut: group 1 has noisier signal
embeddings[250:, 0] += np.random.randn(250) * 2

detector = ShortcutDetector(methods=["equalized_odds"])
detector.fit(embeddings, labels, group_labels=groups)

print(detector.summary())

Expected: Higher TPR/FPR gaps due to differential signal quality across groups.

When to Use Equalized Odds

Use Equalized Odds when:

  • You have binary task labels (required)
  • You have explicit group labels (demographics, environments)
  • You want interpretable fairness metrics
  • You need to report regulatory fairness compliance
  • You want to measure differential error rates across groups

Don't use Equalized Odds when:

  • Labels are multi-class (use other methods)
  • Group labels are unavailable
  • You want unsupervised detection (use HBAC, Geometric)
  • Groups are too small (< 10 samples each)

Theory

Equalized Odds (Hardt et al., 2016) requires:

$$ P(\hat{Y}=1 | Y=y, A=a) = P(\hat{Y}=1 | Y=y, A=a') $$

For all $y \in {0, 1}$ and protected groups $a, a'$.

This decomposes into two constraints:

  • Equal TPR: $P(\hat{Y}=1 | Y=1, A=a) = P(\hat{Y}=1 | Y=1, A=a')$
  • Equal FPR: $P(\hat{Y}=1 | Y=0, A=a) = P(\hat{Y}=1 | Y=0, A=a')$

The detector measures violations as:

$$ \text{TPR Gap} = \max_a \text{TPR}_a - \min_a \text{TPR}_a $$

$$ \text{FPR Gap} = \max_a \text{FPR}_a - \min_a \text{FPR}_a $$

See Also